Q: If $$f(x) = \sqrt{x}$$ and $$g(x) = x^2$ - 9$$, what is the domain of $$(f \circ g)(x)$$?

$$x \leq -3 \text{ or } x \geq 3$$

Question 1

Let $$f(x) = \sqrt{x-1}$$ and $$g(x) = x^2$ + 2. $$What is the domain of the composition $$f \circ g$$?

Accepted Answer

$$x \in \mathbb{R}$$

Question 2

Given $$h(x) = \ln(2x+5)$$, decompose $$h(x) as a $$composition of two functions $$f$$ and $$g$$ such that $$h(x) = f(g(x)). $$Which of the following is a correct decomposition?

Accepted Answer

$$f(x) = \ln(x),\ g(x) = 2x+5$$

Question 3

If $$f(x) = 3x - 4$$ and $$g(x) = x^2$, what is (f$$ \circ g)(x)$$?

Accepted Answer

$$3x^2 - 4$

Question 4

Suppose $$f(x) = \sqrt{x}$$ and $$g(x) = 5 - x. $$What is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x \leq 5$$

Question 5

Given $$h(x) = (2x+1)^3$, which of the following is a valid decomposition h(x) = f(g(x$$))$$?

Accepted Answer

$$f(x) = x^3$,\ g(x) = 2x+1$$

Question 6

Let $$f(x) = \frac{1}{x}$$ and $$g(x) = x-3. $$What is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x 
eq 3$$

Question 7

If $$f(x) = |x|$$ and $$g(x) = x-2$$, what is $$(f \circ g)(x)$$?

Accepted Answer

$$|x-2|$$

Question 8

Given $$h(x) = \sin(4x)$$, which of the following is a correct decomposition $$h(x) = f(g(x))$$?

Accepted Answer

$$f(x) = \sin(x),\ g(x) = 4x$$

Question 9

Let $$f(x) = x^2$ and g(x$$) = \sqrt{x}$$. What is the domain of (f$$ \circ g)(x)$$?

Accepted Answer

$$x \geq 0$$

Question 10

Suppose $$f(x) = \frac{1}{x+2}$$ and $$g(x) = x^2$ - 4. $$What is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x 
eq 2,\ x 
eq -2$$

Question 11

Given $$h(x) = e$$^{3x+1}$$, which of the following is a valid decomposition h(x) = f(g(x$$))$$?

Accepted Answer

$$f(x) = e^x,\ g(x) = 3x+1$$

Question 12

If $$f(x) = \sqrt{x}$$ and $$g(x) = x^2$ - 9$$, what is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x \leq -3 	ext{ or } x \geq 3$$

Question 13

Let $$f(x) = \frac{1}{x}$$ and $$g(x) = \frac{1}{x}. $$What is $$(f \circ g)(x)$$?

Accepted Answer

$$x$$

Question 14

Given $$h(x) = \sqrt{5x-10}$$, which of the following is a correct decomposition $$h(x) = f(g(x))$$?

Accepted Answer

$$f(x) = \sqrt{x},\ g(x) = 5x-10$$

Question 15

Suppose $$f(x) = x^2$ + 1$$ and $$g(x) = 2x. $$What is $$(g \circ f)(x)$$?

Accepted Answer

$$2x^2 + 2$

Question 16

If $$f(x) = \frac{1}{x-1}$$ and $$g(x) = x^2$, what is the domain of (f$$ \circ g)(x)$$?

Accepted Answer

$$x 
eq 1,\ x 
eq -1$$

Question 17

Given $$h(x) = \cos(2x-3)$$, which of the following is a valid decomposition $$h(x) = f(g(x))$$?

Accepted Answer

$$f(x) = \cos(x),\ g(x) = 2x-3$$

Question 18

Let $$f(x) = \sqrt{x+4}$$ and $$g(x) = x-2. $$What is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x \geq -2$$

Question 19

Suppose $$f(x) = x^2$ and g(x) = 3x+1$. What is (f$$ \circ g)(x)$$?

Accepted Answer

$$(3x+1)^2$$

Question 20

Given $$h(x) = \ln(x^2$+1)$$, which of the following is a correct decomposition $$h(x) = f(g(x))$$?

Accepted Answer

$$f(x) = \ln(x),\ g(x) = x^2$+1$$

Guided Practice: Function Composition and Decomposition in College Algebra

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